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solve the Polynomial inequality and express the solution in set notation.
y^2+2y > or = to 4
To solve the polynomial inequality y^2 + 2y ≥ 4, we need to determine the values of y that satisfy the inequality.
Step 1: Begin by setting the inequality to zero by subtracting 4 from both sides of the inequality:
y^2 + 2y - 4 ≥ 0
Step 2: Next, we want to find the values of y that make the expression on the left side of the inequality greater than or equal to zero. We can do this by factoring the quadratic expression or by using the quadratic formula, but in this case, the quadratic doesn't factor nicely, so we'll use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In the equation y^2 + 2y - 4 = 0, a = 1, b = 2, and c = -4.
y = (-(2) ± √((2)^2 - 4(1)(-4))) / (2(1))
y = (-2 ± √(4 + 16)) / 2
y = (-2 ± √(20)) / 2
y = (-2 ± 2√(5)) / 2
y = -1 ± √(5)
Step 3: We have found the two solutions to the equation, which are -1 + √(5) and -1 - √(5). We can now graph the quadratic expression y^2 + 2y - 4 to determine the intervals where the inequality is satisfied.
The graph of the quadratic is a parabola that opens upwards. The x-coordinate of the vertex of this parabola is -1, and since the coefficient of the y^2 term is positive, the parabola opens upwards, indicating that the values of y that make the expression positive are greater than the x-coordinate of the vertex.
Step 4: Now we just need to express the solution in set notation. From the graph, we can see that the inequality is satisfied for y-values greater than or equal to -1 + √(5).
Therefore, the solution in set notation is:
y ≥ -1 + √(5)